What is a 4-connected matroid?

TL;DR

This work develops a framework to extract tangible 4-connectivity from high-order tangles in matroids by introducing the breadth of a tangle via the tangle matroid $M_{\mathcal{T}}$. The main result shows that any tangle of order $k\ge4$ in a matroid $M$ has a weakly $4$-connected minor $N$ with a corresponding tangle $\mathcal{T}'$ of order $k$ whose breadth matches that of the original, and in which $\mathcal{T}'$ is induced by $\mathcal{T}$. This ties the abstract tangle structure to a concrete minor with a robust connectivity property and preserves as much information (breadth) as possible. As a corollary, large $k$-connected sets yield weakly $4$-connected minors that retain the set’s size and connectivity, highlighting an unavoidable-minor phenomenon analogous to graph-theoretic results. The paper further analyzes special cases (order $4$ tangles), boundary cases, and presents examples showing the limits of strengthening beyond weak $4$-connectivity while maintaining breadth, along with discussion and conjectures for future work.

Abstract

The {\em breadth} of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. A matroid $M$ is {\em weakly $4$-connected} if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $λ(X)\geq 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}$ of order $k$, breadth $l$ and has the property that $\mathcal{T}$ is the tangle in $M$ induced by $\mathcal{T}_N$. A set $Z$ of elements of a matroid $M$ is $4$-{\em connected} if $λ(A)\geq\min\{|A\cap Z|,|Z-A|,3\}$ for all $A\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element $4$-connected set where $n\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set.

Theorems & Definitions (121)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6